Fractal fits to Riemann zeros

FractalFitstoRiemannZerosPaulB.SlaterISBER,UniversityofCalifornia,SantaBarbara,CA93106(Dated:February7,2008)1arXiv:math-ph/0606005v226Mar2007AbstractWuandSprung(Phys.Rev.E,48,2595(1993))reproducedtherst500nontrivialRiemannzeros,usingaone-dimensionallocalpotentialmodel.Theyconcluded|andsimilarlyvanZylandHutchinson(Phys.Rev.E,67,066211(2003))|thatthepotentialpossessesafractalstructureofdimensiond=32.Wemodelthenonsmoothuctuatingpartofthepotentialbythealternating-signsineseriesfractalofBerryandLewisA(x;).Settingd=32,weestimatethefrequencyparameter(),plusanoverallscalingparameter()weintroduce.Wesearchforthatpairofparameters(;)whichminimizestheleast-squarestSn(;)ofthelowestneigenvalues|obtainedbysolvingtheone-dimensionalstationary(non-fractal)Schrodingerequationwiththetrialpotential(smoothplusnonsmoothparts)|tothelowestnRiemannzerosforn=25.Fortheadditionalcaseswestudy,n=50and75,wesimplyset=1.ThetsobtainedarecomparedtothosegottenbyusingjustthesmoothpartoftheWu-Sprungpotentialwithoutanyfractalsupplementation.Somelimitedimprovement|5.7261vs.6.39207(n=25),11.2672vs.11.7002(n=50)and16.3119vs.16.6809(n=75)|isfoundinour(non-optimized,computationally-bound)searchprocedures.Theimprovementsarerelativelystronginthevicinitiesof=3and(itssquare)9.Further,weextendtheWu-Sprungsemiclassicalframeworktoincludehigher-ordercorrectionsfromtheRiemann-vonMangoldtformula(beyondtheleading,dominantterm)intothesmoothpotential.PACSnumbers:ValidPACS02.10.De,03.65.Sq,05.45.Df,05.45.MtKeywords:Riemannzeros,Wu-Sprungpotential,Berry-Lewisalternating-signsineseriesfractal,Schrodingerequation,fractalpotential,anescalinglaw,deterministicWeierstrass-Mandelbrotfractalfunc-tion,quantumchaos,Riemann-vonMangoldtformula,rankitnormalitytestElectronicaddress:[e-mail]gtheresultsoftheirpaper,\Riemannzerosandafractalpotential,WuandSprungstatedthat\wehavefoundanalyticallyaone-dimensionallocalpotentialwhichgeneratesthesmoothaverageleveldensityobeyedbytheRiemannzeros.WehavethenshownhowanynitenumberoflowlyingRiemannzeroscanbereproducedbyintroducinguctuationsontopofthepotential.Themysteryofhowaone-dimensionalintegrablesystemcanproducea`chaotic'spectrumisresolvedbyadoptingtheconceptofafractalpotentialwhich,intheinniteNlimit,wouldleadtothesystemhavingadimensionlargerthanone[1,p.2597](cf.[2,3,4,5,6,7,8]).(\Indeed...ndinganHermitianoperatorwhoseeigenvaluesare[theRiemannzeros]maybeimpossiblewithoutintroducingchaoticsystems[2,p.3].)TheWu-SprungpotentialV|whichgeneratesthesmoothaverageleveldensityobeyedbytheRiemannzeros|satisedAbel'sintegralequation[1,eq.(6)],andwaswrittenimplicitlyas[1,eq.(7)](cf.[9][10,sec.4]),xWS(V)=1pVV0lnV02e2+pVlnpV+pVV0pVpVV0:(1)HereV0=3:100739:74123.Ourobjectiveistoreproduce,asbestwecan,theuctuationsontopofthepotentialVWS(x),implicitlygivenby(1),sothattheapplicationoftheSchrodingerequationtotheso-amended(smoothplusfractal)potentialwouldyieldtheRiemannzerosthemselves.Forourexploratorypurposes,weadopt(beingaparticularcaseofadeterministicWeierstrass-Mandelbrot[WM]fractalfunction)thealternating-signsineseriesofBerryandLewis[11,eq.(5)],A(x;)=1m=1(1)msinmx(2d)m;(1d2;1):(2)Here,disthefractaldimension,which|followingthebox-countingargumentofWuandSprung[1](cf.[12])|wetaketobe32.Wehave,inthisd=32Berry-Lewiscontext,aspeciccase,A(x;)=12A(x;);(3)ofthe\anescalinglaw[11,eq.(3)].Wealsoscale|intherst(n=25)ofourthreesetsofanalyses(n=25;50;75)|A(x;)byaparameter,wherenisthenumberofthe3lowestRiemannzerosweaspiretot(sec.IIA).Forthecasesn=50(sec.IIB)and75(sec.IIC),wewillsimplyset=1.Insec.III,wedemonstratehowtoincorporatemoretermsoftheRiemann-vonMangoldtformula[13]forthecumulatednumberofRiemannzerosthanWu-Sprungthemselvesdid,usingasemiclassicalargument,inderivingxWS(V).(Itremains,however,tonumericallyimplementtheselastndings.)II.ANALYSESA.n=25Weproceed,tobegin,tryingtotthersttwenty-veRiemannzerosbyndingdis-tinguishedvaluesofthetwoparameters(and).Werandomlygeneratetrialvalues110and010.(Numerically-speaking,wetruncatethesummationin(2)bysummingfromm=30tom=30(cf.[11,App.]).Wehavenotyetgaugedthesensitivityofthevariousresultsinthispapertothischoiceofcuto|nortothesettingd=32norfurthertothespecicmeasureoft(sum-of-squaredeviations)employed|thoughitwouldcertainlybeofinteresttodosoforanyorallofthem.)Ifweusethesmoothpotentialgivenby(1)itself|withoutanyfractalsupplementation|thesum-of-squaresdeviationofthersttwenty-veeigenvaluesyieldedbyapplicationoftheone-dimensionalstationarySchrodingerequationfromthersttwenty-veRiemannzerosis6.23907(Fig.1).(Thisisonly0.0069percentofthetotal[non-tted]sumofsquaresofthezerosthemselves,thatis,92569.63,soonemightaverthatthesemiclassically-basedsmoothWu-Sprungpotentialisnotablysuccessfulinwell-approximatingtheRiemannzeros.Itis,ofcourse,ourobjectiveheretoreducethissmallpercentageevenfurther.LetusalsonotethatarefereesuggestedthatthescatterinFigs.17and9mightbereducedifthemodulusofthescatterweretobeplotted.)Werandomlygenerated4,007pairsof(;)fromtheindicated